What is the value of \(\rm \int e^x (\cos x - \sin x)dx \)

  1. \(\rm e^x\cdot\cos x + c\)
  2. \(\rm e^x\cdot\sin x + c\)
  3. \(\rm -e^x\cdot\cos x + c\)
  4. \(\rm- e^x\cdot\sin x + c\)

Answer (Detailed Solution Below)

Option 1 : \(\rm e^x\cdot\cos x + c\)
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Detailed Solution

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Concept

\(\rm \int e^x \left(f(x)+f'(x)\right)dx \) = ex f(x) + c

Calculation:

Given: \(\rm \int e^x (\cos x - \sin x)dx \)

Let f(x) = cos x

⇒ f'(x) = - sin x

∴ \(\rm \int e^x (\cos x - \sin x)dx = \rm e^x\cdot \cos x + c\)

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